Abstract
Despite longstanding efforts in the K-12 STEM education community to create meaningful mathematical experiences across disciplines, mathematics continues to be a siloed subject which is tracked based on ability. With an increasing number of high-achieving students enrolling in Algebra I in Grade 7 or earlier, there is a need for research on the readiness of accelerated STEM-intending students to persist in mathematics at the college level. A mathematical mindset framework was employed to explore the relationships between level of Algebra I acceleration, student (n = 2111) and instructor (n = 141) beliefs about sense-making and mistake-making, and attitudes in first-semester college calculus. Findings from a series of multilevel analyses indicate that interactions between student mathematical mindset and perceived progressive teaching practices influence attitudes toward mathematics. While student-centered instruction had a slightly negative effect on attitude, there was a differential effect in relation to student beliefs about sense-making as a metric of success in mathematics. These findings contribute to empirical understandings of mathematical mindset and the complex transition from high school calculus to college calculus. Implications for interdisciplinary STEM education and persistence in STEM undergraduate study are discussed.
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Agustin, M. Z. N., & Agustin, M. A. (2009). Algebra and precalculus skills and performance in first-semester calculus. International Journal of Case Method Research & Application, 21(3), 232–236 http://www.wacra.org/PublicDomain/IJCRA%20xxi_iii_pg232-236%20Agustin.pdf.
American College Testing. (2019). The condition of college & career readiness 2019. https://www.act.org/content/dam/act/unsecured/documents/National-CCCR-2019.pdf
Asante, K. O. (2012). Secondary students' attitudes towards mathematics. IFE Psychologia: An International Journal, 20(1), 121–133 https://www.researchgate.net/publication/236269274_Secondary_students_attitudes_towards_mathematics.
Bardini, C., Pierce, R., Vincent, J., & King, D. (2014). Undergraduate mathematics students' understanding of the concept of function. Journal on Mathematics Education, 5(02), 85–107. https://doi.org/10.22342/jme.5.2.1495.85-107.
Boaler, J. (1997). When even the winners are losers: Evaluating the experiences of top set' students. Journal of Curriculum Studies, 29(2), 165–182. https://doi.org/10.1080/002202797184116.
Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. Jossey-Bass.
Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 172–200). Ablex.
Brahier, D. J. (1995). Mathematical dispositions of students enrolled in first-year algebra. Proceedings of the 17th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH. http://eric.ed.gov/?id=ED389590
Bressoud, D. M. (2017a). Characteristics of successful programs in college calculus [data set]. Mathematical Association of America. https://www.maa.org/programs-and-communities/curriculum%20resources/progress-through-calculus/cspcc-data
Bressoud, D. M. (2017b). Summary report of the workshop on the role of calculus in the transition from high school to college mathematics. https://www.maa.org/sites/default/files/RoleOfCalc_rev.pdf
Bressoud, D. M., Camp, D., & Teague, D. (2012). A joint position statement of the mathematical Association of America and the National Council of teachers of mathematics on teaching calculus. Mathematical Association of America. http://www.maa.org/news/2012_maanctm.html
Bressoud, D. M., Mesa, V., & Rasmussen, C. (2015). Preface. In D. M. Bressoud, V. Mesa, & C. Rasmussen (Eds.), Insights and recommendations from the MAA national study of college calculus. MAA Notes (p. v - viii). Mathematical Association of America. http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf
Carlson, M. P., Madison, B., & West, R. D. (2015). A study of students' readiness to learn calculus. International Journal of Research in Undergraduate Mathematics Education, 1, 209–233. https://doi.org/10.1007/s40753-015-0013-y.
Chen, X. (2015). STEM attrition among high-performing college students in the United States: Scope and potential causes. Journal of Technology and Science Education, 5(1), 41–59. https://doi.org/10.3926/jotse.136.
Clark, M., & Lovric, M. (2008). Suggestion for a theoretical model for secondary-tertiary transition in mathematics. Mathematics Education Research Journal, 20(2), 25–37. https://doi.org/10.1007/BF03217475.
Committee on STEM Education, National Science and Technology Council, the White House. (2018). Charting a course for success: America’s strategy for STEM education. https://www.whitehouse.gov/wp-content/uploads/2018/12/STEM-Education-Strategic-Plan-2018.pdf
De Guzmán, M., Hodgson, B. R., Robert, A., & Villani, V. (1998, August). Difficulties in the passage from secondary to tertiary education. In Proceedings of the International Congress of Mathematicians (Vol. 3, pp. 747-762). Documenta Mathematica. http://www.sbai.uniroma1.it/~giuseppe.accascina/01-Villani/1998-De%20Guzman-Hodgson-Robert-%20Villani-Difficulties_in_Passage_from_Secundary_to_Tertiary_Education.pdf
Domina, T., Hanselman, P., Hwang, N., & McEachin, A. (2016). Detracking and tracking up: Mathematics course placements in California middle schools, 2003-2013. American Educational Research Journal, 53(4), 1229–1266. https://doi.org/10.3102/0002831216650405.
Dweck, C. (2006). Mindset: The new psychology of success. Random House.
Ellis, J., Kelton, M. L., & Rasmussen, C. (2014). Student perceptions of pedagogy and associated persistence in calculus. ZDM International Journal on Mathematics Education, 46(4), 661–673. https://doi.org/10.1007/s11858-014-0577-z.
English, L. D. (2016). STEM education K-12: Perspectives on integration. International Journal of STEM Education, 3(3), 1–6. https://doi.org/10.1186/s40594-016-0036-1.
Ferrini-Mundy, J., & Gaudard, M. (1992). Secondary school calculus: Preparation or pitfall in the study of college calculus? Journal for Research in Mathematics Education, 23, 56–71. https://doi.org/10.2307/749164.
Fitzallen, N. (2015). STEM education: What does mathematics have to offer? Mathematics education in the margins. In M. Marshman, V. Geiger, & A. Bennison (Eds.), Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia (pp. 237–244). https://files.eric.ed.gov/fulltext/ED572451.pdf
Galanti, T. M. (2019). Hyper-acceleration of Algebra I: Narrating opportunity to learn from a situative perspective (publication no. 10289373) [Doctoral dissertation, George Mason University]. ProQuest Dissertations and Theses.
Gamoran, A., & Hannigan, E. C. (2000). Algebra for everyone? Benefits of college-preparatory mathematics for students with diverse abilities in early secondary school. Educational Evaluation and Policy Analysis, 22(3), 241–254. https://doi.org/10.3102/01623737022003241.
Gojak, L. M. (2013, December 3). Algebra: Not “if” but “when”. National Council of Teachers of Mathematics. http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Linda-M_-Gojak/Algebra_-Not-_If_-but-_When_/
Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), 237–254. https://doi.org/10.1007/s10649-007-9100-6.
Hoang, H., Huang, M., Sulcer, B., & Yesilyurt, S. (2017). Carnegie math pathways 2015–2016 impact report: A five-year review. Carnegie Foundation for the Advancement of Teaching. https://www.carnegiefoundation.org/resources/publications/carnegie-math-pathways-2015-2016-impact-report-a-five-year-review/
Hong, Y. Y., Kerr, S., Klymchuk, S., McHardy, J., Murphy, P., Spencer, S., Thomas, M. O. J., & Watson, P. (2009). A comparison of teacher and lecturer perspectives on the transition from secondary to tertiary mathematics education. International Journal of Mathematical Education in Science and Technology, 40(7), 877–889. https://doi.org/10.1080/00207390903223754.
Keller, R. E., Johnson, E., & DeShong, S. (2017). A structural equation model looking at student's participatory behavior and their success in Calculus I. International Journal of STEM Education, 4, 24. https://doi.org/10.1186/s40594-017-0093-0.
Knight, C. (2007). A resilience framework. Health Education, 107(6), 543-555. https://doi.org/10.1108/09654280710827939.
Larson, M. R. (2017). Mathematics learning: A journey, not a sprint. National Council of Teachers of Mathematics. https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Mathematics-Learning_-A-Journey,-Not-a-Sprint/
Lee, C., & Johnston-Wilder, S. (2017). The construct of mathematical resilience. In U. X. Eligio (Ed.), Understanding emotions in mathematical thinking and learning (pp. 269–291). London: Elsevier. https://doi.org/10.1016/B978-0-12-802218-4.00010-8.
Li, Y., & Schoenfeld, A. H. (2019). Problematizing teaching and learning mathematics as “given” in STEM education. International Journal of STEM Education, 6, 44. https://doi.org/10.1186/s40594-019-0197-9.
Li, Y., Schoenfeld, A. H., diSessa, A. A., Graesser, A. C., Bensen, L. C., English, L. D., & Dueschl, R. A. (2019). On thinking and STEM education. Journal for STEM Education Research, 2(1), 1–13. https://doi.org/10.1007/s41979-019-00014-x.
Lipnevich, A. A., Preckel, F., & Krumm, S. (2016). Mathematics attitudes and their unique contribution to achievement: Going over and above cognitive ability and personality. Learning and Individual Differences, 47, 70–79. https://doi.org/10.1016/j.lindif.2015.12.027.
Loveless, T. (2013). Advanced math in eighth grade. The Brookings Institution: Brown Center on Education Policy http://www.brookings.edu/research/reports/2013/03/18-eighth-grade-math-loveless.
Loveless, T. (2016). How well are American students learning? The Brookings Institution: Brown Center on Education Policy https://www.brookings.edu/wp-content/uploads/2016/03/Brown-Center-Report-2016.pdf.
Ma, X., & Willms, J. D. (1999). Dropping out of advanced mathematics: How much do students and schools contribute to the problem? Educational Evaluation and Policy Analysis, 21(4), 365–383. https://doi.org/10.3102/01623737021004365.
Maat, S. M. B., & Zakaria, E. (2010). The learning environment, teacher's factor and students attitude toward mathematics among engineering technology students. International Journal of Academic Research, 2(2), 16–20 https://www.researchgate.net/profile/Effandi_Zakaria/publication/228837745_THE_LEARNING_ENVIRONMENT_TEACHER'S_FACTOR_AND_STUDENTS_ATTITUDE_TOWARDS_MATHEMATICS_AMONGST_ENGINEERING_TECHNOLOGY/links/0912f510c7ee57b694000000.pdf.
Mesa, V., Burn, H., & White, N. J. (2015). Good teaching of Calculus I. In D. M. Bressoud, V. Mesa, & C. L. Rasmussen (Eds.), Insights and recommendations from the MAA National Study of College Calculus. MAA Notes (pp. 83–91). Mathematical Association of America.
Mohamed, L., & Waheed, H. (2011). Secondary students' attitude toward mathematics in a selected school of Maldives. International Journal of Humanities and Social Sciences, 1(15), 277–281 http://www.ijhssnet.com/journals/Vol_1_No_15_Special_Issue_October_2011/34.pdf.
National Council of Teachers of Mathematics. (2018). Catalyzing change in high school mathematics: Initiating critical conversations. National Council of Teachers of Mathematics.
Nicolaidou, M., & Philippou, G. (2003). Attitudes towards mathematics, self-efficacy, and achievement in problem solving. In M. A. Mariottie (Ed.), European research in mathematics education III (pp. 1–11). University of Pisa.
Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students' function understanding. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (Vol. 73, pp. 27–41). Mathematical Association of America Service Center. https://doi.org/10.5948/UPO9780883859759.004.
Pyzdrowski, L. J., Sun, Y., Curtis, R., Miller, D., Winn, G., & Hensel, R. A. (2013). Readiness and attitudes as indicators for success in college calculus. International Journal of Science and Mathematics Education, 11(3), 529–554. https://doi.org/10.1007/s10763-012-9352-1.
Rattan, A., Good, C., & Dweck, C. S. (2012). "It's ok - not everyone can be good at math": Instructors with an entity theory comfort (and demotivate) students. Journal of Experimental Social Psychology, 48(3), 731–737. https://doi.org/10.1016/j.jesp.2011.12.012.
Rickles, J. H. (2013). Examining heterogeneity in the effect of taking algebra in eighth grade. Journal of Educational Research, 106(4), 251–268. https://doi.org/10.1080/00220671.2012.692731.
Sadler, P., & Sonnert, G. (2018). The path to college calculus: The impact of high school mathematics coursework. Journal for Research in Mathematics Education, 49(3), 292-329. https://doi.org/10.5951/jresematheduc.49.3.0292.
Schmidt, W. H. (2009). Exploring the relationship between content coverage and achievement: Unpacking the meaning of tracking in eighth grade mathematics. Education Policy Center at Michigan State University.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of well-taught mathematics courses. Educational Psychologist, 23(2), 145–166. https://doi.org/10.1207/s15326985ep2302_5.
Schommer-Aikins, M., Duell, O. K., & Hutter, R. (2005). Epistemological beliefs, mathematical problem-solving beliefs, and academic performance of middle school students. The Elementary School Journal, 105(3), 289–304. https://doi.org/10.1086/428745.
Schraeder, M. R., Pyzdrowski, L. J., & Miller, D. A. (2019). The impact of prior exposure to calculus. American Journal of Educational Research, 7(3), 237–243. https://doi.org/10.12691/education-7-3-8.
Selden, A. (2005). New developments and trends in tertiary mathematics education: Or, more of the same? International Journal of Mathematical Education in Science and Technology, 36(2–3), 131–147. https://doi.org/10.1080/00207390412331317040.
Singh, K., Granville, M., & Dika, S. (2002). Mathematics and science achievement: Effects of motivation, interest, and academic engagement. Journal of Educational Research, 95, 323–332. https://doi.org/10.1080/00220670209596607.
Smith, J. B. (1996). Does an extra year make any difference? The impact of early access to algebra on long-term gains in mathematics attainment. Educational Evaluation and Policy Analysis, 18(2), 141–153. https://doi.org/10.3102/01623737018002141.
Sonnert, G., Sadler, P. M., Sadler, S. M., & Bressoud, D. M. (2015). The impact of instructor pedagogy on college calculus students' attitude toward mathematics. International Journal of Mathematical Education in Science and Technology, 46(3), 370–387. https://doi.org/10.1080/0020739X.2014.979898.
Stein, M. K., Kaufman, J. H., Sherman, M., & Hillen, A. F. (2011). Algebra: A challenge at the crossroads of policy and practice. Review of Educational Research, 81(4), 453–492. https://doi.org/10.3102/0034654311423025.
Stewart, S., & Reeder, S. (2017). Algebra underperformances at college level: What are the consequences? In S. Stewart (Ed.), And the rest is just algebra (pp. 3–18). Springer International Publishing.
Suh, J. M., Graham, S., Ferranone, T., Kopeinig, G., & Bertholet, B. (2011). Developing persistent and flexible problem solvers with a growth mindset. In D. J. Brahier (Ed.), Motivation and disposition: Pathways to learning mathematics, 73rd yearbook (pp. 169–184). National Council of Teachers of Mathematics.
Sun, K. L. (2018). The role of mathematics teaching in fostering student growth mindset. Journal for Research in Mathematics Education, 49(3), 330–355. https://doi.org/10.5951/jresematheduc.49.3.0330.
Vasquez, J. A., Sneider, M., & Comer, M. (2013). STEM lesson essentials, grades 3–8: Integrating science, technology, engineering and mathematics. Heinemann.
Wade, C., Sonnert, G., Sadler, P., Hazari, Z., & Watson, C. (2016). A comparison of mathematics teachers' and professors' views on secondary preparation for tertiary calculus. Journal of Mathematics Education at Teachers College, 7(1), 7–16 https://journals.library.columbia.edu/index.php/jmetc/article/view/782.
Wang, X. (2013). Why students choose STEM majors: Motivation, high school learning, and postsecondary context of support. American Educational Research Journal, 50, 1081–1121. https://doi.org/10.3102/0002831213488622.
Wilkins, J. L. M., & Ma, X. (2003). Modeling change in student attitude toward beliefs about mathematics. Journal of Educational Research, 97, 52–63. https://doi.org/10.1080/00220670309596628.
Yeager, D. S., & Dweck, C. S. (2012). Mindsets that promote resilience: When students believe that personal characteristics can be developed. Educational Psychologist, 47(4), 302–314. https://doi.org/10.1080/00461520.2012.722805.
Yu, R., & Singh, K. (2018). Teacher support, instructional practices, student motivation, and mathematics achievement in high school. The Journal of Educational Research, 111(1), 81–94. https://doi.org/10.1080/00220671.2016.1204260.
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This material is based upon work supported by the National Science Foundation under grant DRL REESE #0910240. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Galanti, T.M., Miller, A.D. From High School to College Calculus: Beliefs about Sense-Making and Mistakes. Journal for STEM Educ Res 4, 73–94 (2021). https://doi.org/10.1007/s41979-020-00039-7
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DOI: https://doi.org/10.1007/s41979-020-00039-7